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The simple magnifying glass can magnify an image by a relatively small factor – usually no more than about 5. Although larger magnifications are possible, the aberrations of the lens become more important as the magnification increases. The magnifying glass also becomes more and more difficult to use, since the object must be moved closer and closer to the eye as the magnification increases.

The simple magnifier can be combined with other lenses in a number of configurations. Although all of the configurations are similar, each of these configurations is designed for a specific purpose, and the details are different.

The first such configuration is the compound microscope, which is designed to provide greater magnification than can be provided by a single lens. The microscope is designed to magnify objects that can be brought close to the device. The magnified image is inverted, but this is usually not a serious problem for most applications.

In its simplest form, the microscope consists of two positive lenses, which are mounted at either end of a tube. The first lens has a very short focal length, and the object is placed in front of this lens just beyond the focal distance. Since the lens is a positive lens, it forms a real, inverted image at a distance L behind the lens as shown. Since the distance from the first lens to the image is much greater than from the object to the first lens, this image is magnified by the ratio of these two distances. The size of the length L is set by the geometry of the device, and is often about 16 cm.

For example, if the focal length of the first lens is 1 cm and if L is 16 cm, then, using the algebraic form of the ray-tracing rules, the object must be 1.07 cm in front of the lens (that is, just greater than the focal length of 1 cm). The magnification of the first lens is therefore 16/1.07 or about 15. More generally, the magnification of the first lens is approximately L/fo, where fo is the focal length of the objective lens in cm. The magnification could be increased by increasing L, but there is a limit to how big L can become before the microscope tube is too long to be used conveniently.

The second lens uses this real image as its object and acts as a standard simple magnifier. As we showed in the previous section, the magnification of this lens depends somewhat on whether the image it produces is viewed when the eye muscles are relaxed so that the eye lens is focused at infinity or contracted so that the eye lens is focused at the near point. In either case, the magnification is approximately 25/fe, where fe is the focal length of the lens in the eye piece in cm.

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The eye piece lens is often equipped with an adjustment that can move it in and out along the length of the axis of the instrument. This adjustment compensates for the different strengths of the eye lenses of different users. As we showed in the previous section, the combination of the lens in the eye piece of the microscope and the lens in the eye itself must produce a real image on the retina, and the eye lens in the microscope is adjusted until this image is in sharp focus there.

The overall magnification of the microscope is the product of the magnifications of the two lenses, and overall magnifications of 100X are relatively easy to obtain.

The magnification of each lens increases as its focal length is made shorter. Since the f/number is the ratio of the focal length to the diameter, decreasing the focal length also decreases the f/number. A smaller f/number means that the lenses capture more light from the object, which is a clear advantage. However, a smaller f/number also means that the depth of field will be correspondingly reduced. Since the amount of light is usually not a problem in a laboratory environment, the advantage of greater sensitivity is often not very important, whereas any decrease in the depth of field can be serious.

Therefore, the real trade off in the design of a microscope usually is the balance between magnification and depth of field. A common solution to this problem is to decrease both the focal length and the diameter of the objective lens so that the f/number, which is the ratio of these two quantities, remains more or less constant. Most high-magnification objective lenses are very small in diameter for this reason. The idea is to keep the focal length as short as possible to maximize the magnification while keeping the diameter as small as possible to increase the f/number and thereby give the user more depth of field.